Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2

234 Pages·1996·4.38 MB·English

by  J. W. S. Cassels

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES
Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics,
University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England
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86 Topological topics, I.M. JAMES (ed)
87 Surveys in set theory, A.R.D. MATHIAS (ed)
88 FPF ring theory, C. FAITH & S. PAGE
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90 Polytopes and symmetry, S.A. ROBERTSON
92 Representation of rings over skew fields, A.H. SCHOFIELD
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95 Low-dimensional topology 1982, R.A. FENN (ed)
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97 Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN
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99 Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE
100 Stopping time techniques for analysts and probabilists, L. EGGHE
104 Elliptic structures on 3-manifolds, C.B.THOMAS
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107 CompactificationofSiegel moduli schemes, C-L. CHAI
108 Some topics in graph theory, H.P. YAP
109 Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds)
110 An introduction to surreal numbers, H. GONSHOR
113 Lectures on the asymptotic theory of ideals, D. REES
114 Lectures on Bochner-Riesz means, K.M. DAVIS & Y-C. CHANG
115 An introduction to independence for analysts, H.G. DALES & W.H. WOODIN
116 Representations of algebras, P.J.WEBB(ed)
118 Skew linear groups, M. SHIRVANI & B. WEHRFRITZ
119 Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL
121 Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds)
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126 Van der Corput's method of exponential sums, S.W. GRAHAM & G. KOLESNIK
128 Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU
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132 Whitehead groups of finite groups, ROBERT OLIVER
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135 Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds)
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137 Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds)
138 Analysis at Urbana, n, E. BERKSON, T. PECK, & J. UHL (eds)
139 Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds)
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141 Surveys in combinatorics 1989, J. SIEMONS (ed)
144 Introduction to uniform spaces, I.M. JAMES
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148 Helices and vector bundles, A.N. RUDAKOV et al
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150 Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds)
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152 Oligomorphic permutation groups, P.CAMERON
153 L-functions and arithmetic, J. COATES & M.J. TAYLOR (eds)
154 Number theory and cryptography, J. LOXTON (ed
155 Classification theories of polarized varieties, TAKAOFUJITA
156 Twistors in mathematics and physics, T.N. BAILEY & RJ. BASTON (eds)
158 Geometry of Banach spaces, P.F.X. MULLER & W. SCHACHERMAYER (eds)
159 Groups St Andrews 1989 volume 1, CM. CAMPBELL & E.F. ROBERTSON (eds)
160 Groups St Andrews 1989 volume 2, CM. CAMPBELL & E.F. ROBERTSON (eds)
161 Lectures on block theory, BURKHARD KOLSHAMMER
162 Harmonic analysis and representation theory, A. FIGA-TALAMANCA & C. NEBBIA
163 Topics in varieties of group representations, S.M. VOVSI
164 Quasi-symmetric designs, M.S. SHRIKANDE & S.S. SANE
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166 Surveys in combinatorics, 1991, A.D. KEEDWELL (ed)
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174 Lectures on mechanics, J.E. MARSDEN
175 Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds)
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177 Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE, & A. PITTS (eds)
178 Lower K-and L-theory, A.RANICKI
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183 Shintani zeta functions, A. YUKIE
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186 Complexity: knots, colourings and counting, D.J.A. WELSH
187 Surveys in combinatorics, 1993, K. WALKER (ed)
188 Local analysis for the odd order theorem, H. BENDER & G. .GLAUBERMAN
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194 Independent random variables and rearrangement invariant spaces, M. BRAVERMAN
195 Arithmetic of blowup algebras, WOLMER VASCONCELOS
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197 Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI,
W. METZLER & AJ. SIERADSKI (eds)
198 The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN
199 Invariant potential theory in the unit ball of Cn, MANFRED STOLL
200 The Grothendieck theory of dessins d'enfant, L. SCHNEPS (ed)
201 Singularities, JEAN-PAUL BRASSELET (ed)
202 The technique of pseudodifferential operators, H.O. CORDES
203 Hochschild cohomology of von Neumann algebras, A. SINCLAIR & R. SMITH
204 Combinatorial and geometric group theory, AJ. DUNCAN, N.D. GILBERT & J. HOWIE (eds)
205 Ergodic theory and its connections with harmonic analysis, K. PETERSEN & I. SALAMA (eds)
206 An introduction to noncommutative differential geometry and its physical applications, J. MADORE
207 Groups of Lie type and their geometries, W.M. KANTOR & L. DIMARTINO (eds)
208 Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD & W.M. OXBURY (eds)
209 Arithmetic of diagonal hypersurfaces over finite fields, F.Q. GOUVEA & N. YUI
210 Hilbert C*-modules, E.C LANCE
211 Groups 93 Galway / St Andrews I, CM. CAMPBELL et al
212 Groups 93 Galway / St Andrews II, CM. CAMPBELL et al
214 Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders, V. KOWALENKO,
N.E. FRANKEL, M.L. GLASSER & T. TAUCHER
215 Number theory, S. DAVID (ed)
216 Stochastic partial differential equations, A. ETHERIDGE (ed)
217 Quadratic forms with applications to algebraic geometry and topology, A. PFISTER
218 Surveys in combinatorics, 1995, PETER ROWLINSON (ed)
220 Algebraic set theory, A. JOYAL & I. MOERDIJK
221 Harmonic approximation, S.J. GARDINER
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223 Analytic semigroups and semilinear initial boundary value problems, KAZUAKITAIRA
224 Computability, enumerability, unsolvability, S.B. COOPER, T.A. SLAMAN & S.S. WAINER (eds)
226 Novikov conjectures, index theorems and rigidity I, S. FERRY, A. RANICKI & J. ROSENBERG (eds)
227 Novikov conjectures, index theorems and rigidity n, S. FERRY, A. RANICKI & J. ROSENBERG (eds)
228 Ergodic theory of Zd actions, M. POLLICOTT & K. SCHMIDT (eds)
229 Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK
230 Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & E. V. FLYNN
231 Semigroup theory and its applications, K.H. HOFMANN & M.W. MISLOVE (eds)
London Mathematical Society Lecture Note Series. 230
Prolegomena to a Middlebrow
Arithmetic of Curves of Genus 2
J.W.S. Cassels
University of Cambridge
E.V. Flynn
University of Liverpool
1 CAMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS
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Published in the United States of America by Cambridge University Press, New York
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Information on this title: www.cambridge.org/9780521483704
© Cambridge University Press 1996
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1996
A catalogue record for this publication is available from the British Library
ISBN-13 978-0-521 -483 70-4 paperback
ISBN-10 0-521-48370-0 paperback
Transferred to digital printing 2005
Contents
Chapters are divided into sections. Some topics occupy several sections. We
give a name to a section and list it here only when it starts a new topic.
Foreword ix
Background and conventions
0. Introduction xi
1. Algebraically closed ground field xi
2. Perfect ground field xii
Appendix. Finite-dimensional Galois modules xiii
Chapter 1. Curves of genus 2
1. Canonical form 1
2. Group law 3
3. The form is canonical 4
4. Plane quartics 4
Chapter 2. Construction of the jacobian
0. Introduction 6
1. Construction of a basis 6
3. Local behaviour 9
4. Formal group law 10
Appendix I. Blowing up and blowing down 12
Appendix II. Change of coordinates on C 13
Chapter 3. The Kummer surface
0. Introduction 17
1. Construction of the surface. Nodes 18
2. Addition of 2-division points 20
3. Commutativity properties 22
4. The biquadratic forms 23
6. A numerical example 24
7. The tropes 25
8. The jacobian as double cover 26
9. The group law on the jacobian 27
10. Recovery of the curve 29
vi Contents
Chapter 4. The dual of the Kummer
0. Introduction 31
1. Description of Pic3 31
3. /C* is the projective dual of /C 34
4. Tangency 36
5. Explicit dualities 37
Appendix. Rational divisor classes with no rational divisor . . .. 39
Chapter 5. Weddle's surface
0. Introduction 40
1. Symmetroid and Jacobian 41
2. A special case 42
4. Duality 44
Chapter 6. (5/20
0. Introduction 47
1. The homomorphism 48
2. The kernel 50
5. When is 23J C 2(5? 55
6. The Kummer viewpoint 57
8. The norm of p 59
9. A pathology 61
10. The quintic case 61
Chapter 7. The jacobian over local fields. Formal groups
0. Introduction 63
1. Computing the formal group 63
2. General properties of formal groups 67
3. The reduction map 69
4. Torsion in the kernel of reduction 70
5. The order of <3/2<3> when & is a finite extension of <Q> , p ^ oo . 70
p
6. The order of <&/2<& when k = R or C 73
Chapter 8. Torsion
0. Introduction 75
1. Computing the group law 75
2. Computing the torsion of a given jacobian 78
3. Searching for large rational torsion 82
Contents vii
Chapter 9. The isogeny. Theory
0. Introduction 88
4. The Kummer formulation 91
5. Statement of results 92
6. Motivation 93
7. The isogeny 95
8. Generation by radicals 96
9. Consequences for Mordell-Weil 97
10. Isogeny for the curve 98
Chapter 10. The isogeny. Applications
0. Introduction 101
1. The isogeny on the jacobian variety 101
2. An injection on &/(/)(<&) 104
3. Homogeneous spaces J and norm spaces L^ 109
di d2 d2
4. Computing #$/<£(«) • #«/<£(«) when jfc = Q 110
p
Chapter 11. Computing the Mordell-Weil group
0. Introduction 113
1. The Weak Mordell-Weil Theorem 113
2. Performing 2-descent without using homogeneous spaces . . .. 116
3. A worked example of complete 2-descent 121
4. A worked example of descent via isogeny 125
5. Large rank 130
Chapter 12. Heights
0. Introduction 133
1. A height function on (5 133
2. The Mordell-Weil Theorem 137
3. A computational improvement 140
Chapter 13. Rational points. Chabauty's Theorem
0. Introduction 143
1. Chabauty's Theorem 143
2. A worked example 148
Chapter 14. Reducible jacobians
0. Introduction 154
1. The straightforward case 154
2. An awful warning 156
3. The reverse process 157
4. Trying to show that a given jacobian is simple 157
viii Contents
Chapter 15. The endomorphism ring
0. Introduction 160
1. A few examples 160
2. Complex and real multiplication 163
Chapter 16. The desingularized Kummer
0. Introduction 165
1. The surface 165
2. Elementary properties 167
3. Equivalence with /C 168
4. Equivalence with /C* 170
5. Comparison of maps 172
6. Further possible developments 172
Chapter 17. A neoclassical approach
0. Introduction 174
1. Proof of theorem 176
2. The Kummer surface 179
3. A special case 181
4. The construction 183
Chapter 18. Zukunftsmusik
0. Introduction 186
1. Framework 186
2. Principal homogeneous spaces 186
3. Duality 186
4. Canonical height 186
5. Second descents 187
6. Other genera 187
7. A challenge of Serre's 187
Appendix I. MAPLE programs 190
Appendix II. Files available by anonymous ftp 204
Bibliography 207
Index rerum et per sonar um 219
Foreword
The arithmetic (= number theory) of curves of genus 0 is well under-
stood. For genus 1, there is a rich body of theory and conjecture, and in
recent years notable success has been achieved in transforming the latter
into the former. For curves of higher genus there is a rich body of theory
with some spectacular successes (e.g. Faltings' proof of 'Mordell's Conjec-
ture'), but our command is still rudimentary. For genus 1, a concrete ques-
tion about an individual curve can usually be answered by one means or
another: for higher genus, this is far from the case.
For curves of genus 0 and 1 the road between general theory and partic-
ular cases is no one-way street. Numerous individual cases were investigated
by amateurs as well as as by distinguished mathematicians such as Diophan-
tos, Fermat, Euler, Sylvester, Mordell, Selmer, Birch and Swinnerton-Dyer.
Regularities which emerged, sometimes quite unexpectedly, suggested the-
orems, which could sometimes be proved. The new theorems suggested
new questions. For higher genus, existing theory is notoriously unadapted
to the study of individual curves, and few have been elucidated. What is
needed is a corpus of explicit concrete cases and a middlebrow arithmetic
theory which would provide both a practicable means to obtain them and
a framework to understand any unexpected regularities.
Such a theory will, of course, draw freely from the existing body of
knowledge, but the emphasis on feasibility gives new perspectives. Classical
geometers worked over an algebraically closed field, usually the complexes.
Modern geometers take account of the ground field, but regard a field ex-
tension as a cheap manoeuvre. For practical computations it is desperately
expensive: the theory we seek must avoid them at almost all costs.
A natural place to begin is with a curve of genus 2 defined over the ratio-
nals and a natural first problem is to determine its Mordell-Weil group (or
at least the rank). General theory suggests an approach through the (arith-
metic) jacobian, the arithmetic analogue of the complex manifold studied
in the 19th century. Although it figures prominently in the general theory,
we appear to have been the first to describe it explicitly, as a surface in
15-dimensional projective space. It is an inconveniently large object, but
fortunately much information is retained in its Kummer surface. These are
surfaces of degree 4, which were also the subject of intensive 19th century
study over the reals and complexes.